We discuss combinatorial properties of convex sets over arbitrary valued fields. We demonstrate analogs of some classical results for convex sets over the reals, including the fractional Helly theorem and Bárány's theorem on points in many simplices (answering a question of Peterzil and Kaplan). Along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC-dimension. These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field. Joint work with Alex Mennen.